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Sally went to the market to buy some fruit and returned with apples and plums.

She returned home with two times the number of plums than apples. If the plums cost 35 cents each and apples cost 50 cents each, and she spent a total of $12.50 for the fruit which also included the cost of one apple she ate on the way home. How many apples and plums did Sally buy from the market?1) Explain your logic on how you would translate this problem into a system of equations

2) Write the system equations for the problem.

3) Which of the methods you learned in this course would you use to solve the problem, and why?

4) What is your solution to the problem?

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2 Answers

Set the following:

A = number of apples  Sally brought home   

P = number of plums Sally brought home

Sally returned home with two times the number of plums than apples.

So:  P = 2A   

 

Cost of each apple = 50 cents = $0.5

Cost of each plum = 35 cents = $0.35

Total cost = $12.5

Note that the total cost included the cost of one apple Sally ate on her way home.

The number of apples Sally bought at the market is A+1. 

The number of plums Sally bought at the market is P.

(Cost of each apple) * (Number of apples bought)  + (Cost of each plum)  *(Number of apples bought) = Total Cost = $12.5  

0.5*(A+1) + 0.35*P = 12.5    (The $ sign cancelled out on both sides of the equation)

Substitute 2A for P:

0.5*(A+1) + 0.35*(2A) = 12.5

0.5*A + 0.5 + 0.7*A = 12.5

1.2A = 12.5 - 0.5

A = 12/1.2 = 10  (The number of apples Sally brought home) 

The number of apples Sally bought at the market = A + 1 = 10 + 1 = 11

P = 2A = 20  (The Number of plums Sally brought home)

The number of plums Sally bought at the market = P = 2A = 20 

Verify solution:

$0.5*(11) + $0.35*(20) = $5.5 + $7 = $12.5  

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There is more than one way to cut the goose -

Here is another (clearer) way to arrive at the same solution:

A = The number of apples Sally bought at the market 

Cost of each apple is 50 cents = $0.5

P = The number of plums Sally bought at the market

Cost of each plum is 35 cents = $0.35

Sally ate one apple (that she bought) on the way home.

So the number of apples Sally brought home is (A-1). 

The number of plums Sally brought home is still P. 

Sally brought home twice the number of plums than apples.

So the ratio of apples to plums is  P = 2*(A-1)

Total Cost = $0.5*A + $0.35*P = $12.5

After removing the $ sign from both sides of the equation we get 

0.5*A + 0.35*P = 12.5

Or:

0.5A + 0.35P = 12.5

 

Substitute 2(A-1) for P gives

0.5A + 0.35*2(A-1) = 12.5

0.5A + 0.7(A-1) = 12.5

0.5A + 0.7A - 0.7 = 12.5

0.5A + 0.7A = 12.5 + 0.7

1.2A = 13.2

A = 13.2/1.2 = 11

P = 2(A-1) = 2(11-1) = 2(10) =20 

So:

The number of apples Sally bought at the market = 11

The number of plums Sally bought at the market = 20

Verify solution:

$0.5*(11) + $0.35*(20) = $5.5 + $7 = $12.5

 

 

let A be the number of apples brought home, so the number of plums is 2A

.5(A+1)  + .35(2A) = 12.50

the 1 accounts for the eaten apple on the way home

.5A + .5 + .7A = 12.50

.5A + .7A = 12.50 - .5

1.2A = 12.00

A = 10

so Sally bought A + 1 apples or 11 apples and 20 plums.